The formula for the surface area of a sphere is:

`A = 4pir^2`

Note that the answer will be in square units eg cm^{2} or m^{2}, as the answer is an *area*.

The diameter of a sphere is measured as 12cm. What, to 2 decimal places, is the surface area of the sphere?

For a Sphere | Area | `= 4pir^2` |

Substitute (`r=frac(1)(2) xx 12`) | `A` | `= pi(6)^2` |

Divide both sides by 100 | `A` | `= 113.0973` |

Answer: 113.10 cm^{2}

An experimental package is to be lowered into the sea to investigate global warming at the bottom of the ocean. Part of the experiment includes a hemisphere which is to be covered in a special coating. The hemisphere is solid, is of a constant thickness throughout and has no holes or apertures. The coating costs £4.50 per square centimetre.

The hemisphere has an external diameter of 40cm and has a constant thickness of 5mm. If the hemisphere has to be totally covered, what will the cost of the coating be?

Total Surface Area = Surface area outside + surface area inside + rim

Radius of outside = 40 ÷ 2 = 20cm

Radius of inside = 20 - 5 = 15cm

Area Sphere_{outside} |
Area | `= 4pir^2` |

Substitute | `A` | `= 4pi(20)^2` |

`= 5026.55` | ||

Area Sphere_{inside} |
Area | `= 4pir^2` |

Substitute | `= 4pi(15)^2` | |

`= 2827.43` | ||

Area Rim_{outer} |
Area | `= pir^2` |

Substitute | `= pi(20)^2` | |

`= 1256.64` | ||

Area Rim_{inner} |
Area | `= pir^2` |

Substitute | `= pi(15)^2` | |

`= 706.86` |

Total Surface Area

= 5026.55 + 2827.43 + (1256.64 - 706.86)

= 8403.76

Total Cost = 8403.76 x £4.50 = £37,816.92

Answer: £37816.92

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